I just bought recordings of The Boys From Syracuse and Babes in Arms. From earlier I have a recording of Pal Joey. What other Rodgers and Hart shows are considered essential and have good recordings of them?

You already have Pal Joey and Babes in Arms, so I'd say that the only essential show score you're missing is On Your Toes, which I'd say is only second to Pal Joey in terms of its significance in the development of musical theatre (solely in regard to Richard Rodgers and Lorenz Hart's contributions, naturally). It might be a little simplistic to reduce Oklahoma! to an equation consisting of Pal Joey, On Your Toes (created by Richard Rodgers) and Show Boat (created by Oscar Hammerstein II), but those three shows really established all of the major elements made popular by Oklahoma! when Rodgers and Hammerstein finally collaborated on the creation of a Broadway musical.

The following recordings of On Your Toes are available: 1954 Revivaland 1983 Revival. I own the 1983 revival, although I must admit hardly ever listen to it. Still, it is nice to have.

The other shows are obviously incredibly interesting for the wealth of popular song material that originated from the scores, much of which has been highlighted in the thread above. Many are worth having, I'd say, but they are not perhaps what some might label "essentials", and they could be added as and when one can afford them. The pair's songbook is amazing and I'm somewhat surprised it hasn't been tapped for a high profile "revisal" for one of the shows in the way that Crazy For You was cerated out of the bare bones of Girl Crazy and the Gershwin songbook.